A combinatorial approach to the diagonal $N$-representability problem
Mark Laurance
Yoseloff
1-41
Abstract: The problem considered is that of the diagonal N-representability of a pth-order reduced density matrix, $p \geq 2$, for a system of N identical fermions or bosons. A finite number M of allowable single particle states is assumed. The problem is divided into three cases, namely: Case I. $M = N + p$ ; Case II. $M < N + p$; Case III. $M > N + p$. Using the theory of polyhedral convex cones, a complete set of necessary and sufficient conditions is first found for Case I. This solution is then employed to find such conditions for Case II. For Case III, two algorithms are developed to generate solutions for the problem, and examples of the usage of these algorithms are given.
Classical quotient rings
Robert C.
Shock
43-48
Abstract: Throughout R is a ring with right singular ideal $ Z(R)$. A right ideal K of R is rationally closed if $ {x^{ - 1}}K = \{ y \in R:xy \in K\}$ is not a dense right ideal for all $x \in R - K$. A ring R is a Cl-ring if R is (Goldie) right finite dimensional, $ R/Z(R)$ is semiprime, $ Z(R)$ is rationally closed, and $Z(R)$ contains no closed uniform right ideals. These rings include the quasi-Frobenius rings as well as the semiprime Goldie rings. The commutative Cl-rings have Cl-classical quotient rings. The injective ones are congenerator rings. In what follows, R is a Cl-ring. A dense right ideal of R contains a right nonzero divisor. If R satisfies the minimum condition on rationally closed right ideals then R has a classical Artinian quotient ring. The complete right quotient ring Q (also called the Johnson-Utumi maximal quotient ring) of R is a Cl-ring. If R has the additional property that bR is dense whenever b is a right nonzero divisor, then Q is classical. If Q is injective, then Q is classical.
A bounded difference property for classes of Banach-valued functions
Wilbur P.
Veith
49-56
Abstract: Let $A(G,E)$ denote the set of functions f from a Hausdorff topological group G to a Banach space E such that the range of f is relatively compact in E and $\phi \circ f$ is in $A(G,C)$ for each $\phi$ in the dual of E, where $ A(G,C)$ is a translation-invariant ${C^\ast}$ algebra of bounded, continuous, complex-valued functions on G with respect to the supremum norm and complex conjugation. $A(G,E)$ has the bounded difference property if whenever $F:G \to E$ is a bounded function such that ${\Delta _t}F(x) = F(tx) - F(x)$ is in $ A(G,E)$ for each t in G, then F is also an element of $ A(G,E)$. A condition on $ A(G,C)$ and a condition on E are given under which $A(G,E)$ has the bounded difference property. The condition on $A(G,C)$ is satisfied by both the class of almost periodic functions and the class of almost automorphic functions.
Square integrable representations of semisimple Lie groups
Juan A.
Tirao
57-75
Abstract: Let D be a bounded symmetric domain. Let G be the universal covering group of the identity component ${A_0}(D)$ of the group of all holomorphic diffeomorphisms of D onto itself. In this case, any G-homogeneous vector bundle $E \to D$ admits a natural structure of G-homogeneous holomorphic vector bundles. The vector bundle $E \to D$ must be holomorphically trivial, since D is a Stein manifold. We exhibit explicitly a holomorphic trivialization of $E \to D$ by defining a map $ \Phi :G \to {\text{GL}}(V)$ (V being the fiber of the vector bundle) which extends the classical ``universal factor of automorphy'' for the action of ${A_0}(D)$ on D. Then, we study the space H of all square integrable holomorphic sections of $E \to D$. The natural action of G on H defines a unitary irreducible representation of G. The representations obtained in this way are square integrable over $ G/Z$ (Z denotes the center of G) in the sense that the absolute values of their matrix coefficients are in ${L_2}(G/Z)$.
Indecomposable polytopes
Walter
Meyer
77-86
Abstract: The space of summands (with respect to vector addition) of a convex polytope in n dimensions is studied. This space is shown to be isomorphic to a convex pointed cone in Euclidean space. The extreme rays of this cone correspond to similarity classes of indecomposable polytopes. The decomposition of a polytope is described and a bound is given for the number of indecomposable summands needed. A means of determining indecomposability from the equations of the bounding hyperplanes is given.
Goldie-like conditions on Jordan matrix rings
Daniel J.
Britten
87-98
Abstract: In this paper Goldie-like conditions are put on a Jordan matrix ring $J = H({R_n},{\gamma _a})$ which are necessary and sufficient for R to be a $\ast$-prime Goldie ring or a Cayley-Dickson ring. Existing theory is then used to obtain a Jordan ring of quotients for J.
Class numbers of real quadratic number fields
Ezra
Brown
99-107
Abstract: This article is a study of congruence conditions, modulo powers of two, on class number of real quadratic number fields $Q(\sqrt d )$, for which d has at most three distinct prime divisors. Techniques used are those associated with Gaussian composition of binary quadratic forms.
Injective modules and localization in noncommutative Noetherian rings
Arun Vinayak
Jategaonkar
109-123
Abstract: Let $\mathfrak{S}$ be a semiprime ideal in a right Noetherian ring R and $ \mathcal{C}(\mathfrak{S}) = \{ c \in R\vert[c + \mathfrak{S}$ regular in $R/\mathfrak{S}\} $. We investigate the following two conditions: $ ({\text{A}})\;\mathcal{C}(\mathfrak{S})$ is a right Ore set in R. $ ({\text{B}})\;\mathcal{C}(\mathfrak{S})$ is a right Ore set in R and the right ideals of $ {R_{\mathfrak{S}}}$, the classical right quotient ring of R w.r.t. $ \mathcal{C}(\mathfrak{S})$ are closed in the $ J({R_{\mathfrak{S}}})$-adic topology. The main results show that conditions (A) and (B) can be characterized in terms of the injective hull of the right R-module $R/\mathfrak{S}$. The J-adic completion of a semilocal right Noetherian ring is also considered.
Primitive satisfaction and equational problems for lattices and other algebras
Kirby A.
Baker
125-150
Abstract: This paper presents a general method of solving equational problems in all equational classes of algebras whose congruence lattices are distributive, such as those consisting of lattices, relation algebras, cylindric algebras, orthomodular lattices, lattice-ordered rings, lattice-ordered groups, Heyting algebras, other lattice-ordered algebras, implication algebras, arithmetic rings, and arithmetical algebras.
An algebraic property of the \v Cech cohomology groups which prevents local connectivity and movability
James
Keesling
151-162
Abstract: Let C denote the category of compact Hausdorff spaces and $H:C \to HC$ be the homotopy functor. Let $S:C \to SC$ be the functor of shape in the sense of Holsztyński for the projection functor H. Let X be a continuum and ${H^n}(X)$ denote n-dimensional Čech cohomology with integer coefficients. Let ${A_x} = {\text{char}}\;{H^1}(X)$ be the character group of ${H^1}(X)$ considering ${H^1}(X)$ as a discrete group. In this paper it is shown that there is a shape morphism $F \in {\text{Mor}_{SC}}(X,{A_X})$ such that ${F^\ast}:{H^1}({A_X}) \to {H^1}(X)$ is an isomorphism. It follows from the results of a previous paper by the author that there is a continuous mapping $f:X \to {A_X}$ such that $S(f) = F$ and thus that ${f^\ast}:{H^1}({A_X}) \to {H^1}(X)$ is an isomorphism. This result is applied to show that if X is locally connected, then ${H^1}(X)$ has property L. Examples are given to show that X may be locally connected and $ {H^n}(X)$ not have property L for $n > 1$. The result is also applied to compact connected topological groups. In the last section of the paper it is shown that if X is compact and movable, then for every integer n, $ {H^n}(X)/{\operatorname{Tor}}\;{H^n}(X)$ has property L. This result allows us to construct peano continua which are nonmovable. An example is given to show that ${H^n}(X)$ itself may not have property L even if X is a finite-dimensional movable continuum.
Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients
Jean-Pierre
Gossez
163-205
Abstract: Variational boundary value problems for quasilinear elliptic systems in divergence form are studied in the case where the nonlinearities are nonpolynomial. Monotonicity methods are used to derive several existence theorems which generalize the basic results of Browder and Leray-Lions. Some features of the mappings of monotone type which arise here are that they act in nonreflexive Banach spaces, that they are unbounded and not everywhere defined, and that their inverse is also unbounded and not everywhere defined.
Koebe sequences of arcs and normal meromorphic functions
Stephen
Dragosh
207-222
Abstract: Let f be a normal meromorphic function in the unit disk. An estimate for the growth of the modulus of f on a Koebe sequence of arcs is obtained; the estimate is in terms of the order of normality of f. An immediate consequence of the estimate is the following theorem due to F. Bagemihl and W. Seidel: A nonconstant normal meromorphic function has no Koebe values. Another consequence is that each level set of a nonconstant normal meromorphic function cannot contain a Koebe sequence of arcs provided the order of normality of f is less than a certain positive constant ${C^\ast}$.
A Siegel formula for orthogonal groups over a function field
Stephen J.
Haris
223-231
Abstract: We obtain a Siegel formula for a quadratic form over a function field, by establishing the convergence of the corresponding Eisenstein-Siegel series directly, then via the Hasse principle, that of the associated Poisson formula.
Conditions for a TVS to be homeomorphic with its countable product
Wesley E.
Terry
233-242
Abstract: C. Bessaga has given conditions for a Banach space to be homeomorphic with its countable product. In this paper, we extend and generalize these results to complete metric topological vector spaces by using infinite dimensional techniques.
A new characterization of tame $2$-spheres in $E\sp{3}$
Lawrence R.
Weill
243-252
Abstract: It is shown in Theorem 1 that a 2-sphere S in ${E^3}$ is tame from $A = {\text{Int}}\;S$ if and only if for each compact set $F \subset A$ there exists a 2-sphere $ S'$ with complementary domains $A' = {\text{Int}}\;S',B' = {\text{Ext}}\;S'$, such that $F \subset A' \subset \overline {A'} \subset A$ and for each $x \in S'$ there exists a path in $\rho (F,S)$ which runs from x to a point $ y \in S$. Furthermore, the theorem holds when A is replaced by B, $ A'$ by $B',B'$ by $A'$, and Int by Ext. Two applications of this characterization are given. Theorem 2 states that a 2-sphere is tame from the complementary domain C if for arbitrarily small $ \varepsilon > 0$, S has a metric $ \varepsilon$-envelope in C which is a 2-sphere. Theorem 3 answers affirmatively the following question: Is a 2-sphere $S \subset {E^3}$ tame in ${E^3}$ if there exists an $\varepsilon > 0$ such that if a, $ b \in S$ satisfy $\rho (a,b) < \varepsilon$, then there exists a path in S of spherical diameter $\rho (a,b)$ which connects a and b?
Standard polynomials in matrix algebras
Louis H.
Rowen
253-284
Abstract: Let ${M_n}(F)$ be an $n \times n$ matrix ring with entries in the field F, and let ${S_k}({X_1}, \ldots ,{X_k})$ be the standard polynomial in k variables. Amitsur-Levitzki have shown that ${S_{2n}}({X_1}, \ldots ,{X_{2n}})$ vanishes for all specializations of ${X_1}, \ldots ,{X_{2n}}$ to elements of $ {M_n}(F)$. Now, with respect to the transpose, let $ M_n^ - (F)$ be the set of antisymmetric elements and let $M_n^ + (F)$ be the set of symmetric elements. Kostant has shown using Lie group theory that for n even ${S_{2n - 2}}({X_1}, \ldots ,{X_{2n - 2}})$ vanishes for all specializations of ${X_1}, \ldots ,{X_{2n - 2}}$ to elements of $M_n^ - (F)$. By strictly elementary methods we have obtained the following strengthening of Kostant's theorem: ${S_{2n - 2}}({X_1}, \ldots ,{X_{2n - 2}})$ vanishes for all specializations of ${X_1}, \ldots ,{X_{2n - 2}}$ to elements of $M_n^ - (F)$, for all n. ${S_{2n - 1}}({X_1}, \ldots ,{X_{2n - 1}})$ vanishes for all specializations of ${X_1}, \ldots ,{X_{2n - 2}}$ to elements of $M_n^ - (F)$ and of ${X_{2n - 1}}$ to an element of $M_n^ + (F)$, for all n. $ {S_{2n - 2}}({X_1}, \ldots ,{X_{2n - 2}})$ vanishes for all specializations of $ {X_1}, \ldots ,{X_{2n - 3}}$ to elements of $ M_n^ - (F)$ and of ${X_{2n - 2}}$ to an element of $M_n^ + (F)$, for n odd. These are the best possible results if F has characteristic 0; a complete analysis of the problem is also given if F has characteristic 2.
On dynamical systems with the specification property
Karl
Sigmund
285-299
Abstract: A continuous transformation T of a compact metric space X satisfies the specification property if one can approximate distinct pieces of orbits by single periodic orbits with a certain uniformity. There are many examples of such transformations which have recently been studied in ergodic theory and statistical mechanics. This paper investigates the relation between Tinvariant measures and the frequencies of T-orbits. In particular, it is shown that every invariant measure (and even every closed connected subset of such measures) has generic points, but that the set of all generic points is of first category in X. This generalizes number theoretic results concerning decimal expansions and normal numbers.
Ergodic measure preserving transformations with quasi-discrete spectrum
James B.
Robertson
301-311
Abstract: It is shown that an ergodic measure preserving transformation with quasi-discrete spectrum is conjugate to: (a) the skew-product of an ergodic measure preserving transformation with discrete spectrum and a measurable family of totally ergodic measure preserving transformations with quasi-discrete spectrum; (b) a factor of the direct product of an ergodic measure preserving transformation with discrete spectrum and a totally ergodic measure preserving transformation with quasi-discrete spectrum. Sufficient conditions are given to insure that an ergodic measure preserving transformation with quasi-discrete spectrum is conjugate to the direct product of an ergodic measure preserving transformation with discrete spectrum and a totally ergodic measure preserving transformation with quasi-discrete spectrum.
Convolution operators on $G$-holomorphic functions in infinite dimensions
Philip J.
Boland;
Seán
Dineen
313-323
Abstract: For a complex vector space E, let ${H_G}(E)$ denote the space of G (Gateaux)-holomorphic functions on $E\;(f:E \to C$ is G-holomorphic if the restriction of f to every finite dimensional subspace of E is holomorphic in the usual sense). The most natural topology on ${H_G}(E)$ is that of uniform convergence on finite dimensional compact subsets of E. A convolution operator A on ${H_G}(E)$ is a continuous linear mapping $ A:{H_G}(E) \to {H_G}(E)$ such that A commutes with translations. The concept of a convolution operator generalizes that of a differential operator with constant coefficients. We prove that if A is a convolution operator on $ {H_G}(E)$, then the kernel of A is the closed linear span of the exponential polynomials contained in the kernel. In addition, we show that any nonzero convolution operator on $ {H_G}(E)$ is a surjective mapping.
Finite extensions of minimal transformation groups
Robert J.
Sacker;
George R.
Sell
325-334
Abstract: In this paper we shall study homomorphisms $p:W \to Y$ on minimal transformation groups. We shall prove, in the case that W and Y are metrizable, that W is a finite (N-to-1) extension of Y if and only if W is an N-fold covering space of Y and p is a covering map. This result places no further restrictions on the acting group. We shall then use this characterization to investigate the question of lifting an equicontinuous structure from Y to W. We show that, under very weak restrictions on the acting group, this lifting is always possible when W is a finite extension of Y.
Singular perturbations for systems of linear partial differential equations
A.
Livne;
Z.
Schuss
335-343
Abstract: We consider the system of linear partial differential equations $ \varepsilon {A^{ij}}u_{ij}^\varepsilon + {B^i}u_i^\varepsilon + C{u^\varepsilon } = f$ where ${A^{ij}},{B^i}$ are symmetric $m \times m$ matrices and -- C is a sufficiently large positive definite matrix. We prove that under suitable conditions ${\left\Vert {{u^\varepsilon } - u} \right\Vert _{{L^2}}} \leq c\surd \varepsilon {\left\Vert f \right\Vert _{{H^1}}}$ where u is the solution of a suitable boundary value problem for the system ${B^i}{u_i} + Cu = f$.
Property $SUV\sp{\infty }$ and proper shape theory
R. B.
Sher
345-356
Abstract: A class of spaces called the $ SU{V^\infty }$ spaces has arisen in the study of a possibly noncompact variant of cellularity. These spaces play a role in this new theory analogous to that of the $ U{V^\infty }$ spaces in cellularity theory. Herein it is shown that the locally compact metric space X is an $SU{V^\infty }$ space if and only if there exists a tree T such that X and T have the same proper shape. This result is then used to classify the proper shapes of the $SU{V^\infty }$ spaces, two such being shown to have the same proper shape if and only if their end-sets are homeomorphic. Also, a possibly noncompact analog of property $U{V^n}$, called $SU{V^n}$, is defined and it is shown that if X is a closed connected subset of a piecewise linear n-manifold, then X is an $ SU{V^n}$ space if and only if X is an $ SU{V^\infty }$ space. Finally, it is shown that a locally finite connected simplicial complex is an $ SU{V^\infty }$ space if and only if all of its homotopy and proper homotopy groups vanish.
Global dimension of tiled orders over commutative noetherian domains
Vasanti A.
Jategaonkar
357-374
Abstract: Let R be a commutative noetherian domain and $ \Lambda = ({\Lambda _{ij}}) \subseteq {M_n}(R)$ be a tiled R-order. The main result of this paper is the following Theorem. Let gl $\dim R = d < \infty$ and $\Lambda$ a triangular tiled R-order (i.e., $ {\Lambda _{ij}} = R$ whenever $i \leq j$). Then the following three conditions are equivalent: (1) gl $\dim \Lambda < \infty $; (2) ${\Lambda _{i,i - 1}} = R$ or gl $ \dim \;(R/{\Lambda _{i,i - 1}}) < \infty$, whenever $2 \leq i \leq n$; (3) gl $\dim \Lambda \leq d(n - 1)$. If $d = 1$ or 2 then the upper bound in the above theorem is best possible. We give a sufficient condition for an arbitrary tiled R-order $\Lambda$ to be of finite global dimension.
Semigroups with midunits
Janet E.
Ault
375-384
Abstract: A semigroup has a midunit u if $aub = ab$ for all a and b. It is the purpose of this paper to explore semigroups with midunits, and to find an analog to the group of units. In addition, two constructions will be given which produce some semigroups with midunits, and an abstract characterization will be made of certain of these semigroups.
Countable unions of totally projective groups
Paul
Hill
385-391
Abstract: Let the p-primary abelian group G be the set-theoretic union of a countable collection of isotype subgroups $ {H_n}$ of countable length. We prove that if ${H_n}$ is totally projective for each n, then G must be totally projective. In particular, an ascending sequence of isotype and totally projective subgroups of countable length leads to a totally projective group. The result generalizes and complements a number of theorems appearing in various articles in the recent literature. Several applications of the main result are presented.
On structure spaces of ideals in rings of continuous functions
David
Rudd
393-403
Abstract: A ring of continuous functions is a ring of the form $C(X)$, the ring of all continuous real-valued functions on a completely regular Hausdorff space X. With each ideal I of $ C(X)$, we associate certain subalgebras of $C(X)$, and discuss their structure spaces. We give necessary and sufficient conditions for two ideals in rings of continuous functions to have homeomorphic structure spaces.
Isometries of $\sp{\ast} $-invariant subspaces
Arthur
Lubin
405-415
Abstract: We consider families of increasing $^\ast$-invariant subspaces of ${H^2}(D)$, and from these we construct canonical isometrics from certain ${L^2}$ spaces to ${H^2}$. We give necessary and sufficient conditions for these maps to be unitary, and discuss the relevance to a problem concerning a concrete model theory for a certain class of operators.
Regular self-injective rings with a polynomial identity
Efraim P.
Armendariz;
Stuart A.
Steinberg
417-425
Abstract: This paper studies maximal quotient rings of semiprime P. I.-rings; such rings are regular, self-injective and satisfy a polynomial identity. We show that the center of a regular self-injective ring is regular self-injective; this enables us to establish that the center of the maximal quotient ring of a semiprime P. I.-ring R is the maximal quotient ring of the center of R, as well as some other relationships. We give two decompositions of a regular self-injective ring with a polynomial identity which enable us to show that such rings are biregular and are finitely generated projective modules over their center.
Erratum to: ``Weakly almost periodic functionals on the Fourier algebra'' (Trans. Amer. Math. Soc. {\bf 185} (1973), 501--514)
Charles F.
Dunkl;
Donald E.
Ramirez
427